EE 205
Coifman
Composite Waveforms- combining the basic waveforms to
make really nasty stuff.
e.g, express the following curve in terms of simple waveforms:
v(t)
VA
0.632V A
0.5 V A
0
0
TC
2T C
16-1
3T C
4T C
5T C
t
EE 205
Coifman
Putting the hand on the other foot, characterize the
following waveform:
v( t ) =
r (t )
VAe − t TC u(t )
TC
[
]
v(t)
0.4 V A
0.3 V A
0.2 V A
0.1 V A
0
0
TC
2T C
3T C
16-2
4T C
5T C
t
EE 205
Coifman
And you thought you only had two feet... characterize this
waveform too:
[
]
v(t ) = sin ω ot VAe − t TC u(t )
v(t)
VA
-t/T C
VA e
0.5 VA
t
0
TC
3T C
2T C
- 0.5 VA
T0
- VA
16-3
4T C
EE 205
Coifman
And back to stepping on our hands, develop an equation
for the following square wave:
v(t)
T0
VA
t
-V A
16-4
EE 205
Coifman
Back to the books...
instantaneous value vs. waveform/function/signal
Temporal descriptors:
a signal is periodic iff ∃ To s.t. v(t+To)=v(t) ∀ t
a signal is causal iff ∃ T s.t. v(t)≡0 ∀ t<T
(If you can’t follow this notation, the book says the same thing in English)
16-5
EE 205
Coifman
Amplitude descriptors:
When we’re out together dancing peek to peek...
peek to peek voltage (or other signal) is defined by:
Vpp = VMax − VMin
peek voltage (or other signal) is defined by:
Vp = max( VMax , −VMin )
v(t)
V MAX
t
0
V MIN
v(t)
V MAX
V MIN
t
0
16-6
EE 205
Coifman
average voltage (or other signal) over interval T is defined
by:
Vavg
1
=
T
t +T
∫ v( x )dx
t
v(t)
Areas are equal
VA
V avg
+
+
0
+
2T 0
T0
3T 0
t
v(t)
Areas are equal
VA
+
+
V avg
+
t
-
T0
-
- VA
16-7
2T 0
-
3T 0
EE 205
Coifman
Root mean square (RMS) - applying what we know to
something you will see much more of in future classes
instantaneous power: p(t ) =
average power:
Pavg
1
=
T
1
2
v(t )]
[
R
t +T
1 1
∫t p( x )dx = R  T

t +T

∫t [v( x )] dx 

2
now take the square root and note that "mean"="avarage"
RMS voltage:
finally,
Pavg =
Vrms =
1
T
t +T
2
v
x
dx
(
)
[
]
∫
t
1 2
Vrms
R
RMS voltage provides a measure of the power carried by
a periodic signal.
16-8
EE 205
Coifman
"If you put a tin can in the microwave, you’ll be as powerful
as they are"
v(t ) = VA cos(ω Ot + φ )
this periodic signal is characterized by three parameters:
Amplitude: VA
Frequency: ωo
Phase: φ
By combining sinusoids (some times an infinite number of
them), we can build any periodic signal.
In fact, if the signal of interest has a frequency ωo (how
does this relate to T?) we will only need to use sinusoids
at the fundamental frequency, ωo and harmonic
frequencies n·ωo.
16-9
Amplitude
50
50
50
50
0
50
0
0
2
0
0
0
2
0
0
-50
2
-50
0
-50
2
-50
0
30
1
t
30
1
t
30
20
1
t
30
20
1
t
30
20
0
0
20
5
20
w
10
0
0
10
5
10
w
10
0
0
10
5
2
w
2
1
0
0
2
1
5
2
1
0
0
2
1
-2
0
1
5
0
w
0
-2
0
0
5
0
w
0
-2
0
-1
5
-1
w
-1
-2
0
-1
5
-1
-2
0
w
w
-50
Amplitude
Phase (rad/π)
1
t
w
w
2
5
5
Amplitude
Amplitude
Phase (rad/π)
1
1
1
0
1
-1
1
2
0.5
1
0.5
0
0.5
-1
0.5
2
0.5
1
0
0
0
-1
0
2
0
1
0
0
-0.5
-1
-0.5
2
-0.5
1
t
-0.5
0
t
-0.5
-1
t
1
t
1
1
0
0
1
5
1
0
0
1
5
0.5
0
0
0.5
5
1
w
0.5
0
0
1
w
0.5
5
w
0.5
0
0
1
0.5
w
0.5
1
0.5
-1
0
0.5
5
0.5
w
0
-1
0
0
5
0
w
0
-1
0
0
5
-0.5
w
-0.5
-1
0
-0.5
5
-0.5
w
-0.5
-1
0
1
t
w
w
5
5
2